5.3 - MATH
1081
 Wednesday,
March
2
 


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Unformatted text preview: MATH
1081
 Wednesday,
March
2
 
 Chapter
5
–
Section
3
 
 HIGHER
DERIVATIVES,
CONCAVITY,
AND
 THE
SECOND
DERIVATIVE
TEST
 
 Homework
#7
(due
3/7)
 







 
 Section
5.2
#30,
36,
46,
56
 Section
5.3
#30,
44,
52,
78,
82,
94
 Clicker
Check­In:

Choose
any
letter
to
check
in
now.
 To
recap
what
we
have
been
doing
in
the
last
two
sections,
we
 are
using
the
sign
of
the
derivative
to
determine
where
a
 function
is
increasing
and
where
it
is
decreasing.

More
 precisely,
on
intervals
where
 f ' ( x ) > 0 ,
the
function
 f 
is
 increasing
and
on
intervals
where
 f ' ( x ) < 0 ,
the
function
 f 
is
 decreasing.
 
 We
can
then
use
that
information
to
determine
the
location
of
 relative
extrema,
looking
at
points
in
the
domain
where
the
 function
 f 
changes
direction.
 
 This
analysis
gives
us
a
rough
idea
of
what
the
graph
of
the
 function
must
look
like.

However,
this
analysis
alone
leaves
 some
information
about
the
shape
of
the
graph
unknown.

For
 example,
just
looking
at
the
sign
of
the
derivative,
there
is
no
 difference
between
the
two
functions
shown
here.
 
 
 However,
there
clearly
is
a
difference
between
the
shapes
of
 these
functions.
 
 Consider
these
two
curves,
both
of
which
are
increasing
on
the
 interval
( 0, 5 ) .
 
 
 

 
 
 How
can
we
describe
the
difference
between
the
shapes
of
 these
curves?
 What
can
we
say
about
the
tangent
lines
in
each
case? Now
consider
these
two
curves,
both
of
which
are
decreasing
 on
the
interval
( 0, 5 ) .
 
 
 
 
 
 How
can
we
describe
the
difference
between
the
shapes
of
 these
curves?
 What
can
we
say
about
the
tangent
lines
in
each
case?
 Definition:
Concavity
 A
function
is
concave
up
on
an
interval
( a, b )
if
the
graph
of
 the
function
lies
above
its
tangent
line
at
each
point
in
the
 interval.
 A
function
is
concave
down
on
an
interval
( a, b ) 
if
the
graph
 of
the
function
lies
below
its
tangent
line
at
each
point
in
 the
interval.
 A
point
at
which
the
graph
changes
concavity
is
called
an
 inflection
point.
 
 From
the
discussion
on
the
last
two
slides,
“concave
up”
 corresponds
to
a
curve
where
the
slopes
of
the
tangent
lines
 are
increasing,
whereas,
“concave
down”
corresponds
to
a
 curve
where
the
slopes
of
the
tangent
lines
are
decreasing.
 
From
Section
5.1,
we
have
a
method
to
determine
where
a
 function
is
increasing
and
where
something
is
decreasing…we
 look
at
its
derivative.
 
 So,
to
measure
the
concavity
of
a
function
 f ,
we
need
to
 analyze
the
derivative
of
its
“slope
function”.

But
the
“slope
 function”
of
 f 
is
 f ' ( x ) .

This
means
that
we
want
to
analyze
the
 derivative
of
the
derivative
of
 f ,
which
we
call
the
second
 derivative
of
 f .
 
 Notation:
Common
notations
for
the
second
derivative
are
 
 d2y 2 


 
 
 Dx ⎡ f ( x ) ⎤ 
 f '' ( x ) 
 

 
 ⎣ ⎦ dx 2 Example:

Find
the
second
derivative
for
the
function.
 
 
 
 
 1.
 f ( x ) = −7 x 2 + 9 x + 3
 
 
 
 
 8 
 
 
 2.

 g ( x ) = 5 x 4 / 3 − 
 x 
 
 
 
 5x − x2 
 
 
 3.

 y = e − 2 e 
 Suppose
 s ( t ) 
gives
us
the
position
of
an
object
at
time
 t .
 
 What
physical
interpretation
does
the
derivative
 s ' ( t ) 
have?
 
 
 What
physical
interpretation
does
the
second
derivative
 s '' ( t ) 
 then
have? Given
an
algebraic
expression
for
a
function
 f ( x ) ,
we
must
 look
at
the
sign
of
the
second
derivative
 f '' ( x ) 
(because
that
 will
tell
us
if
 f ' ( x ) 
is
increasing
or
decreasing)
to
determine
 concavity.
 
 If
both
 f ' 
and
 f '' 
exist
at
all
points
in
an
interval
( a, b ) 
and

 
 • f '' ( x ) > 0 
at
all
 x 
in
that
interval,
then
 f 
is
concave
upward
 on
( a, b ) .
 • f '' ( x ) < 0 
at
all
 x 
in
that
interval,
then
 f 
is
concave
 downward
on
( a, b ) .
 
 A
point
in
the
domain
of
 f 
where
the
concavity
(sign
of
 f '' )
 changes
is
an
inflection
point.
 Example:

Determine
the
intervals
on
which
the
function
is

 concave
upward
and
the
intervals
on
which
the



 function
is
concave
downward
and
identify
any



 inflection
points.
 
 
 
 
 1.

 f ( x ) = −7 x 2 + 9 x + 3
 
 
 5 
 
 
 2.

 g ( x ) = 
 x−2 
 
 
 − x2 
 
 
 3.

 h ( x ) = 2 e 
 One
useful
application
of
the
point
of
inflection
is
to
describe
 the
point
of
diminishing
return.
 
 Suppose
you
begin
a
new
ad
campaign.

At
first,
there
is
an
 explosion
in
your
customer
base…the
rate
of
increase
of
the
 number
of
customers
is
high.
 
 Eventually,
the
rate
at
which
new
customers
appear
will
slow
 down.

You
are
still
adding
new
customers
(so
the
first
 derivative
is
still
positive),
but
at
a
slower
rate
(so
the
second
 derivative
turns
negative).
 

 Clicker
Check­out:

Choose
any
letter
to
check
out
now.
 
 
 Tomorrow
in
recitation:

Workshop
#4
 
 Next
time
–
Section
6.1
 ...
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